In this work we solved the equation of scalar and tensor perturbations for the generalized Starob... more In this work we solved the equation of scalar and tensor perturbations for the generalized Starobinsky inflationary model using the improved uniform approximation method and the phase-integral method up to third-order in deviation. We compare our results with the numerical integration. We have obtained that both semiclassical methods reproduce the scalar power spectra P S,T , the scalar spectral index n S , and the tensor-to-scalar ratio r. Also we present our results in the (n S , r) plane.
The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological... more The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological perturbations in the quartic chaotic inflationary model. The phaseintegral formulas for the scalar power spectrum are explicitly obtained up to fifth order of the phase-integral approximation. As in previous reports [1-3], we point out that the accuracy of the phase-integral approximation compares favorably with the numerical results and those obtained using the slow-roll and uniform approximation methods.
We present a new potential barrier that presents the phenomenon of superradiance when the reflect... more We present a new potential barrier that presents the phenomenon of superradiance when the reflection coefficient [Formula: see text] is greater than one. We calculated the transmission and reflection coefficients for three different regions. The results are compared with those obtained for the hyperbolic tangent potential barrier and the step potential barrier. We also present the solution of the Klein–Gordon equation with the Lambert-[Formula: see text] potential barrier in terms of the Heun Confluent functions.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service... more This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights Analytical solution to the equation of motion of original (numerically solved) model. We explore effects of several parameters on evolution of the instability. Delayed fluidization time models forcing & granular response temporal coupling. Model now describes abrupt changes observed at large vibration accelerations.
We solve the Klein–Gordon equation for a step potential with hyperbolic tangent potential. The sc... more We solve the Klein–Gordon equation for a step potential with hyperbolic tangent potential. The scattering solutions are derived in terms of hypergeometric functions. The reflection coefficient R and transmission coefficient T are calculated, we observed superradiance and transmission resonances.
We solve the Klein–Gordon equation in the presence of the hyperbolic tangent potential. The scatt... more We solve the Klein–Gordon equation in the presence of the hyperbolic tangent potential. The scattering solutions are derived in terms of hypergeometric functions. The reflection, R, and transmission, T, coefficients are calculated in terms of gamma function and, superradiance is discussed, when the reflection coefficient, R, is greater than one.
We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth p... more We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker [Formula: see text] function, and the antiparticle bound state is discussed in terms of potential parameters.
We present a study of the one-dimensional Klein–Gordon equation by a smooth barrier. The scatteri... more We present a study of the one-dimensional Klein–Gordon equation by a smooth barrier. The scattering solutions are given in terms of the Whittaker Mκ,μ(x) function. The reflection and transmission coefficients are calculated in terms of the energy, the height, and the smoothness of the potential barrier. For any value of the smoothness parameter we observed transmission resonances.
Recently a versatile methodology for the evaluation of the mixed flocculation-coalescence rate (k... more Recently a versatile methodology for the evaluation of the mixed flocculation-coalescence rate (kFC) of oil-in-water (o/w) nanoemulsions was proposed [Rahn-Chique et al. (2012), Interciencia 37: 582-587]. The value of kFC is obtained fitting a theoretical expression of the turbidity of the emulsion (τTEO) to its observed behavior (τEXP) over a short period of time. The expression of τTEO contains an additional parameter which was originally introduced to account for the fraction of collisions between the drops that lead to their aggregation (xa). Hence, under some approximations, the value of (1-xa) should represent the fraction of collisions that leads to the coalescence of the drops. Here, two sets of ionic nanoemulsions, and two sets of latex particles are used to study the dependence of xa with the average radius of the emulsion. The results suggest that despite its original justification, (1-xa) is more likely to represent the contribution of the “spherical” scattering of light...
Emulsion Stability Simulations (ESS) of deformable droplets are used to study the influence of th... more Emulsion Stability Simulations (ESS) of deformable droplets are used to study the influence of the time-dependent adsorption on the coalescence time of a 200-µm drop of soybean oil pressed by buoyancy against a planar water/oil interface. The interface is represented by a 5000-µm drop of oil fixed in the space. The movement of the small drop is determined by the interaction forces between the drops, the buoyancy force, and its thermal interaction with the solvent. The interaction forces depend on the surface concentration of surfactant molecules at the oil/water interfaces. Assuming diffusion limited adsorption, the surface excess of the surfactant becomes a function of its apparent diffusion constant, D app. Distinct probability distributions of the coalescence time are obtained depending on the magnitude of D app. The origin and the significance of these distributions are discussed.
In a previous report [C. Rojas, G. Urbina-Villalba, M. García-Sucre, Phys. Rev. E 81, 016302 (201... more In a previous report [C. Rojas, G. Urbina-Villalba, M. García-Sucre, Phys. Rev. E 81, 016302 (2010)] it was shown that Emulsion Stability Simulations (ESS) are able to reproduce the lifetime of micrometer-size drops of hexadecane pressed by buoyancy against a planar water/hexadecane interface. It was confirmed that small drops (ri < 10 µm) stabilized with β−casein behave as non-deformable particles, moving with a combination of Stokes and Taylor tensors as they approach the interface. Here, a similar methodology is used to parametrize the potential of interaction of drops of soybean oil stabilized with Bovine Serum Albumin (BSA). The potential obtained is then employed to study the lifetime of deformable drops in the range 10 µm ≤ ri ≤ 1000 µm. It is established that the average lifetime of these drops can be adequately replicated using the model of truncated spheres. However, the results depend sensibly on the expressions of the initial distance of deformation and the maximum film radius used in the calculations. The set of equations adequate for large drops is not satisfactory for medium-size drops (10 µm ≤ ri ≤ 100 µm.) and vice versa. In the case of large particles, the increase of the interfacial area as a consequence of the deformation of the drops generates a very large repulsive barrier which opposes coalescence. Nevertheless, the buoyancy force prevails. As a consequence, it is the hydrodynamic tensor of the drops which determine the characteristic behavior of the lifetime as a function of the particle size. While the average values of the coalescence time of the drops can be justified by the mechanism of film thinning, the scattering of the experimental data of large drops cannot be rationalized using the methodology previously described. A possible explanation of this phenomenon required elaborate simulations which combine deformable drops, capillary waves, repulsive interaction forces, and a time-dependent surfactant adsorption.
The coalescence of liquid drops induces a higher level of complexity compared to the classical st... more The coalescence of liquid drops induces a higher level of complexity compared to the classical studies about the aggregation of solid spheres. Yet, it is commonly believed that most findings on solid dispersions are directly applicable to liquid mixtures. Here, the state of the art in the evaluation of the flocculation rate of these two systems is reviewed. Special emphasis is made on the differences between suspensions and emulsions. In the case of suspensions, the stability ratio is commonly evaluated from the initial slope of the absorbance as a function of time under diffusive and reactive conditions. Puertas and de las Nieves (1997) developed a theoretical approach that allows the determination of the flocculation rate from the variation of the turbidity of a sample as a function of time. Here, suitable modifications of the experimental procedure and the referred theoretical approach are implemented in order to calculate the values of the stability ratio and the flocculation rate corresponding to a dodecane-in-water nanoemulsion stabilized with sodium dodecyl sulfate. Four analytical expressions of the turbidity are tested, basically differing in the optical cross section of the aggregates formed. The first two models consider the processes of: a) aggregation (as described by Smoluchowski) and b) the instantaneous coalescence upon flocculation. The other two models account for the simultaneous occurrence of flocculation and coalescence. The latter reproduce the temporal variation of the turbidity in all cases studied (380 ≤ [NaCl] ≤ 600 mM), providing a method of appraisal of the flocculation rate in nanoemulsions.
In this work, we calculate the tensor power spectrum and the tensor-to-scalar ratio [Formula: see... more In this work, we calculate the tensor power spectrum and the tensor-to-scalar ratio [Formula: see text] within the frame of the Starobinsky inflationary model using the improved uniform approximation method and the third-order phase-integral method. We compare our results with those obtained with numerical integration and the slow-roll approximation to second-order. We have obtained consistent values of [Formula: see text] using the different approximations, and [Formula: see text] is inside the interval reported by observations.
arXiv: General Relativity and Quantum Cosmology, 2020
In this paper we present the study of the scalar cosmological perturbations of a single field inf... more In this paper we present the study of the scalar cosmological perturbations of a single field inflationary model up to first order in deviation. The Christoffel symbols and the tensorial quantities are calculated explicitly in function of the cosmic time t. The Einstein equations are solved up-to first order in deviation and the scalar perturbations equation is derived.
In this work we study the scalar power spectrum and the spectral index for the Starobinsky inflat... more In this work we study the scalar power spectrum and the spectral index for the Starobinsky inflationary model using the phase integral method up-to third-order of approximation. We show that the semiclassical methods reproduce the scalar power spectrum for the Starobinsky model with a good accuracy, and the value of the spectral index compares favorably with observations. Also, we compare the results with the uniform approximation method and the second-order slow-roll approximation.
Computation of the Power Spectrum in Chaotic $1/4 \lambda \phi^4$ Inflation
The phase-integral approximation devised by Fr\"oman and Fr\"oman, is used for computin... more The phase-integral approximation devised by Fr\"oman and Fr\"oman, is used for computing cosmological perturbations in the quartic chaotic inflationary model. The phase-integral formulas for the scalar power spectrum are explicitly obtained up to fifth order of the phase-integral approximation. As in previous reports [1-3], we point out that the accuracy of the phase-integral approximation compares favorably with the numerical results and those obtained using the slow-roll and uniform approximation methods.
We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential.... more We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential. The scattering solutions are obtained in terms of Whittaker functions and the condition for the existence of transmission resonances is derived. We show the dependence of the zero-reflection condition on the shape of the potential. In the low momentum limit, transmission resonances are associated with half-bound states. We express the condition for transmission resonances in terms of the phase shifts.
The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological... more The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological perturbations in the quadratic chaotic inflationary model. The phase-integral formulas for the scalar and tensor power spectra are explicitly obtained up to fifth order of the phase-integral approximation. We show that, the phase integral gives a very good approximation for the shape of the power spectra associated with scalar and tensor perturbations as well as the spectral indices. We find that the accuracy of the phase-integral approximation compares favorably with the numerical results and those obtained using the slow-roll and uniform approximation methods.
In this work we solved the equation of scalar and tensor perturbations for the generalized Starob... more In this work we solved the equation of scalar and tensor perturbations for the generalized Starobinsky inflationary model using the improved uniform approximation method and the phase-integral method up to third-order in deviation. We compare our results with the numerical integration. We have obtained that both semiclassical methods reproduce the scalar power spectra P S,T , the scalar spectral index n S , and the tensor-to-scalar ratio r. Also we present our results in the (n S , r) plane.
The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological... more The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological perturbations in the quartic chaotic inflationary model. The phaseintegral formulas for the scalar power spectrum are explicitly obtained up to fifth order of the phase-integral approximation. As in previous reports [1-3], we point out that the accuracy of the phase-integral approximation compares favorably with the numerical results and those obtained using the slow-roll and uniform approximation methods.
We present a new potential barrier that presents the phenomenon of superradiance when the reflect... more We present a new potential barrier that presents the phenomenon of superradiance when the reflection coefficient [Formula: see text] is greater than one. We calculated the transmission and reflection coefficients for three different regions. The results are compared with those obtained for the hyperbolic tangent potential barrier and the step potential barrier. We also present the solution of the Klein–Gordon equation with the Lambert-[Formula: see text] potential barrier in terms of the Heun Confluent functions.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service... more This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights Analytical solution to the equation of motion of original (numerically solved) model. We explore effects of several parameters on evolution of the instability. Delayed fluidization time models forcing & granular response temporal coupling. Model now describes abrupt changes observed at large vibration accelerations.
We solve the Klein–Gordon equation for a step potential with hyperbolic tangent potential. The sc... more We solve the Klein–Gordon equation for a step potential with hyperbolic tangent potential. The scattering solutions are derived in terms of hypergeometric functions. The reflection coefficient R and transmission coefficient T are calculated, we observed superradiance and transmission resonances.
We solve the Klein–Gordon equation in the presence of the hyperbolic tangent potential. The scatt... more We solve the Klein–Gordon equation in the presence of the hyperbolic tangent potential. The scattering solutions are derived in terms of hypergeometric functions. The reflection, R, and transmission, T, coefficients are calculated in terms of gamma function and, superradiance is discussed, when the reflection coefficient, R, is greater than one.
We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth p... more We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker [Formula: see text] function, and the antiparticle bound state is discussed in terms of potential parameters.
We present a study of the one-dimensional Klein–Gordon equation by a smooth barrier. The scatteri... more We present a study of the one-dimensional Klein–Gordon equation by a smooth barrier. The scattering solutions are given in terms of the Whittaker Mκ,μ(x) function. The reflection and transmission coefficients are calculated in terms of the energy, the height, and the smoothness of the potential barrier. For any value of the smoothness parameter we observed transmission resonances.
Recently a versatile methodology for the evaluation of the mixed flocculation-coalescence rate (k... more Recently a versatile methodology for the evaluation of the mixed flocculation-coalescence rate (kFC) of oil-in-water (o/w) nanoemulsions was proposed [Rahn-Chique et al. (2012), Interciencia 37: 582-587]. The value of kFC is obtained fitting a theoretical expression of the turbidity of the emulsion (τTEO) to its observed behavior (τEXP) over a short period of time. The expression of τTEO contains an additional parameter which was originally introduced to account for the fraction of collisions between the drops that lead to their aggregation (xa). Hence, under some approximations, the value of (1-xa) should represent the fraction of collisions that leads to the coalescence of the drops. Here, two sets of ionic nanoemulsions, and two sets of latex particles are used to study the dependence of xa with the average radius of the emulsion. The results suggest that despite its original justification, (1-xa) is more likely to represent the contribution of the “spherical” scattering of light...
Emulsion Stability Simulations (ESS) of deformable droplets are used to study the influence of th... more Emulsion Stability Simulations (ESS) of deformable droplets are used to study the influence of the time-dependent adsorption on the coalescence time of a 200-µm drop of soybean oil pressed by buoyancy against a planar water/oil interface. The interface is represented by a 5000-µm drop of oil fixed in the space. The movement of the small drop is determined by the interaction forces between the drops, the buoyancy force, and its thermal interaction with the solvent. The interaction forces depend on the surface concentration of surfactant molecules at the oil/water interfaces. Assuming diffusion limited adsorption, the surface excess of the surfactant becomes a function of its apparent diffusion constant, D app. Distinct probability distributions of the coalescence time are obtained depending on the magnitude of D app. The origin and the significance of these distributions are discussed.
In a previous report [C. Rojas, G. Urbina-Villalba, M. García-Sucre, Phys. Rev. E 81, 016302 (201... more In a previous report [C. Rojas, G. Urbina-Villalba, M. García-Sucre, Phys. Rev. E 81, 016302 (2010)] it was shown that Emulsion Stability Simulations (ESS) are able to reproduce the lifetime of micrometer-size drops of hexadecane pressed by buoyancy against a planar water/hexadecane interface. It was confirmed that small drops (ri < 10 µm) stabilized with β−casein behave as non-deformable particles, moving with a combination of Stokes and Taylor tensors as they approach the interface. Here, a similar methodology is used to parametrize the potential of interaction of drops of soybean oil stabilized with Bovine Serum Albumin (BSA). The potential obtained is then employed to study the lifetime of deformable drops in the range 10 µm ≤ ri ≤ 1000 µm. It is established that the average lifetime of these drops can be adequately replicated using the model of truncated spheres. However, the results depend sensibly on the expressions of the initial distance of deformation and the maximum film radius used in the calculations. The set of equations adequate for large drops is not satisfactory for medium-size drops (10 µm ≤ ri ≤ 100 µm.) and vice versa. In the case of large particles, the increase of the interfacial area as a consequence of the deformation of the drops generates a very large repulsive barrier which opposes coalescence. Nevertheless, the buoyancy force prevails. As a consequence, it is the hydrodynamic tensor of the drops which determine the characteristic behavior of the lifetime as a function of the particle size. While the average values of the coalescence time of the drops can be justified by the mechanism of film thinning, the scattering of the experimental data of large drops cannot be rationalized using the methodology previously described. A possible explanation of this phenomenon required elaborate simulations which combine deformable drops, capillary waves, repulsive interaction forces, and a time-dependent surfactant adsorption.
The coalescence of liquid drops induces a higher level of complexity compared to the classical st... more The coalescence of liquid drops induces a higher level of complexity compared to the classical studies about the aggregation of solid spheres. Yet, it is commonly believed that most findings on solid dispersions are directly applicable to liquid mixtures. Here, the state of the art in the evaluation of the flocculation rate of these two systems is reviewed. Special emphasis is made on the differences between suspensions and emulsions. In the case of suspensions, the stability ratio is commonly evaluated from the initial slope of the absorbance as a function of time under diffusive and reactive conditions. Puertas and de las Nieves (1997) developed a theoretical approach that allows the determination of the flocculation rate from the variation of the turbidity of a sample as a function of time. Here, suitable modifications of the experimental procedure and the referred theoretical approach are implemented in order to calculate the values of the stability ratio and the flocculation rate corresponding to a dodecane-in-water nanoemulsion stabilized with sodium dodecyl sulfate. Four analytical expressions of the turbidity are tested, basically differing in the optical cross section of the aggregates formed. The first two models consider the processes of: a) aggregation (as described by Smoluchowski) and b) the instantaneous coalescence upon flocculation. The other two models account for the simultaneous occurrence of flocculation and coalescence. The latter reproduce the temporal variation of the turbidity in all cases studied (380 ≤ [NaCl] ≤ 600 mM), providing a method of appraisal of the flocculation rate in nanoemulsions.
In this work, we calculate the tensor power spectrum and the tensor-to-scalar ratio [Formula: see... more In this work, we calculate the tensor power spectrum and the tensor-to-scalar ratio [Formula: see text] within the frame of the Starobinsky inflationary model using the improved uniform approximation method and the third-order phase-integral method. We compare our results with those obtained with numerical integration and the slow-roll approximation to second-order. We have obtained consistent values of [Formula: see text] using the different approximations, and [Formula: see text] is inside the interval reported by observations.
arXiv: General Relativity and Quantum Cosmology, 2020
In this paper we present the study of the scalar cosmological perturbations of a single field inf... more In this paper we present the study of the scalar cosmological perturbations of a single field inflationary model up to first order in deviation. The Christoffel symbols and the tensorial quantities are calculated explicitly in function of the cosmic time t. The Einstein equations are solved up-to first order in deviation and the scalar perturbations equation is derived.
In this work we study the scalar power spectrum and the spectral index for the Starobinsky inflat... more In this work we study the scalar power spectrum and the spectral index for the Starobinsky inflationary model using the phase integral method up-to third-order of approximation. We show that the semiclassical methods reproduce the scalar power spectrum for the Starobinsky model with a good accuracy, and the value of the spectral index compares favorably with observations. Also, we compare the results with the uniform approximation method and the second-order slow-roll approximation.
Computation of the Power Spectrum in Chaotic $1/4 \lambda \phi^4$ Inflation
The phase-integral approximation devised by Fr\"oman and Fr\"oman, is used for computin... more The phase-integral approximation devised by Fr\"oman and Fr\"oman, is used for computing cosmological perturbations in the quartic chaotic inflationary model. The phase-integral formulas for the scalar power spectrum are explicitly obtained up to fifth order of the phase-integral approximation. As in previous reports [1-3], we point out that the accuracy of the phase-integral approximation compares favorably with the numerical results and those obtained using the slow-roll and uniform approximation methods.
We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential.... more We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential. The scattering solutions are obtained in terms of Whittaker functions and the condition for the existence of transmission resonances is derived. We show the dependence of the zero-reflection condition on the shape of the potential. In the low momentum limit, transmission resonances are associated with half-bound states. We express the condition for transmission resonances in terms of the phase shifts.
The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological... more The phase-integral approximation devised by Fröman and Fröman, is used for computing cosmological perturbations in the quadratic chaotic inflationary model. The phase-integral formulas for the scalar and tensor power spectra are explicitly obtained up to fifth order of the phase-integral approximation. We show that, the phase integral gives a very good approximation for the shape of the power spectra associated with scalar and tensor perturbations as well as the spectral indices. We find that the accuracy of the phase-integral approximation compares favorably with the numerical results and those obtained using the slow-roll and uniform approximation methods.
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Papers by Clara Rojas